A simple definition for locally compact quantum groups

In this Note we propose a simple definition of a locally compact quantum gr oup in reduced form. By the word "reduced" we mean that we suppose the Haar weight to be faithful, and hence we define in fact arbitrary locally compa ct quantum groups represented on the L-2-space of the Haar weight. We const ruct the multiplicative unitary associated with our quantum group. We const ruct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontrya gin duality theorem. (C) Academie des Sciences/Elsevier, Paris.